Two-stage low-complexity max-log bit-level llr calculator and method

ABSTRACT

A demodulator and demodulation method includes a bit/symbol hard demodulator configured to obtain hard bit or symbol information from a received signal. At least one lookup table is configured to reference coefficients for computation of log-likelihood ratios (LLRs) from the hard bit or symbol information. A log-likelihood ratio calculation module is configured to compute bit-level LLRs from the coefficients and the received signal.

BACKGROUND

1. Technical Field

The present invention relates to modulation and demodulation ofinformation and more particularly to a two-stage low-complexitytable-driven Max-Log bit-by-bit log-likelihood ratio (LLR) calculatorand method.

2. Description of the Related Art

In many modern communication systems, where a soft decoder is employed,it is necessary to calculate a soft estimate for data bits from receivedsignals. A maximum a posteriori (MAP) demodulator can be used togenerate the bit-by-bit log-likelihood ratio (LLR), this is too complexfor the practical implementation.

To avoid the ultra-high complexity of the MAP algorithm, Max-Log ispractically used to compute the LLR, e.g., it is included in the 3GPPstandard (Section A.1.4 of 3GPP TS 25.848). However, the complexity ofMax-Log is high even for a constellation of moderate size, say, 64-QAM(Quadrature Amplitude Modulation) if no further simplification isconsidered.

For QAM constellations, based on the observation that a properlydesigned QAM constellation can be decomposed into two PAM (PulseAmplitude Modulation) constellations, (see e.g., X. Gu, K. Niu, and W.Wu, “A novel efficient soft output demodulation algorithm for high ordermodulation,” in Proc. 4th Int. Conf. Computer Inform. Technol., Wuhan,China, September 2004, pp. 493-498 (hereinafter Gu et al.). This greatlyreduces the complexity.

For PSK constellations, K. Fagervik and T. G. Jeans, in “flow complexitybit by bit soft output demodulator,” Electron. Lett., vol. 32, pp.985-987, May 1999 (hereinafter Fagervik et al.), proposed an approximateMax-Log, which is less complex compared to the exact Max-Log withnegligible performance loss. However, the complexity of the abovestate-of-the-art demodulators is still very high if high-ordermodulations are used, say, 1024-QAM and 32-PSK. The algorithms in Gu etal. and Fagervik et al. require many operations such as multiplications,additions, and comparisons to obtain results.

SUMMARY

A demodulator and demodulation method includes a bit/symbol harddemodulator configured to obtain hard bit or symbol information from areceived signal. At least one lookup table is configured to referencecoefficients for computation of log-likelihood ratios (LLRs) from thehard bit or symbol information. A log-likelihood ratio calculationmodule is configured to compute bit-level LLRs from the coefficients andthe received signal.

These and other features and advantages will become apparent from thefollowing detailed description of illustrative embodiments thereof,which is to be read in connection with the accompanying drawings.

BRIEF DESCRIPTION OF DRAWINGS

The disclosure will provide details in the following description ofpreferred embodiments with reference to the following figures wherein:

FIG. 1 is a block/flow diagram showing a two-stage low-complexitymax-log bit-level LLR calculator and method in accordance with oneillustrative embodiment;

FIG. 2 is an irregular binary reflected Gray Code labeling method inaccordance with one embodiment;

FIG. 3 is a regular binary reflected Gray Code labeling method inaccordance with one embodiment;

FIG. 4 is an illustrative unit circle employed in mapping hard bits to aPSK-8 constellation in accordance with one embodiment;

FIG. 5 is a block/flow diagram showing a system/method for a demodulatorfor performing bit-level LLR in accordance with the present principles.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

In accordance with the present principles, a novel modulator/demodulatorsystem, device and method are provided. In one particularly usefulembodiment, a two-stage low-complexity table-driven Max-Log bit-by-bitlog-likelihood ratio (LLR) calculator is devised for both QuadratureAmplitude Modulation (QAM) and Pulse Amplitude Modulation (PSK)constellations by exploiting the Gray-code structure. A method inaccordance with the present principles reduces the implementationcomplexity dramatically while offering improved performance.

The present embodiments provide much lower complexity compared to priorart demodulators. Specifically, present embodiments, based on lookuptables, only need one multiplication and at most three additions tocompute the LLR per bit for any QAM modulation order, and about threemultiplications and two additions for the practical PSK modulationorder. For QAM constellations, the present embodiments can reduce thecomputational complexity up to, e.g., 81.25%, 83.33%, 91.67% and 95.83%in 16-, 64-, 256- and 1024-QAM, respectively, compared to theapproximate Max-Log algorithm of Fagervik et al. while keeping the sameperformance.

For PSK constellations, the present embodiments can save the complexityup to 71.74%, 75.90% and 81.58% in 8-, 16- and 32-PSK, respectively,compared to the PSK demodulator of Gu et al. while providing betterperformance.

The present embodiments advantageously can be applied immediately incurrent and future communication systems.

Embodiments described herein may be entirely hardware, entirely softwareor including both hardware and software elements. In a preferredembodiment, the present invention is implemented in software, whichincludes but is not limited to firmware, resident software, microcode,etc.

Embodiments may include a computer program product accessible from acomputer-usable or computer-readable medium providing program code foruse by or in connection with a computer or any instruction executionsystem. A computer-usable or computer readable medium may include anyapparatus that stores, communicates, propagates, or transports theprogram for use by or in connection with the instruction executionsystem, apparatus, or device. The medium can be magnetic, optical,electronic, electromagnetic, infrared, or semiconductor system (orapparatus or device) or a propagation medium. The medium may include acomputer-readable medium such as a semiconductor or solid state memory,magnetic tape, a removable computer diskette, a random access memory(RAM), a read-only memory (ROM), a rigid magnetic disk and an opticaldisk, etc.

Notations: we reserve j=√{square root over (−1)} for the imaginary unit,∥·∥ for the amplitude of a complex number, |·| for the absolute value ofa real number, {circumflex over (x)} for the estimate of x, and kε[M,N]denotes that k is an integer with M≦k≦N.

Referring now to the drawings in which like numerals represent the sameor similar elements and initially to FIG. 1, a block/flow diagram showsa system/method 30 in accordance with one embodiment. Two stages areshown (10 and 22). In a 1^(st) stage 10, we obtain the bit- orsymbol-level hard demodulation 14 from a received signal 12 in a veryefficient way. In a 2^(nd) stage 22, we retrieve coefficients from alookup table 16, and compute LLRs 20 for all bits with an LLR calculator18. More details will be given herein on the efficient method ofdemodulator 14 to obtain the bit- and symbol-level hard demodulation;the lookup table 16 for QAM and PSK constellation; and the computation18 of the bit-level LLR 20 using the receive signal and the coefficientsfrom the lookup table 16.

System Model: The system model in one embodiment may be given by {tildeover (z)}={tilde over (x)}+{tilde over (e)}, where {tilde over(z)}=z₁+jz_(Q), {tilde over (x)}=x₁+jx_(Q), is the unnormalizedtransmitted symbol, {tilde over (e)}=e₁+je_(Q) is the additive noisewith power E[∥{tilde over (e)}∥²]=γ⁻¹. The un-normalized 2^(2N)-QAMconstellation is given by

X={2i−1−2^(N) +j(2q−1−2),i,qε[1,2^(N)]}.

Signal Constellation Construction and Modulation; 2^(2N)-QAMConstellation Construction and Modulation: A 2^(2N)-QAM constellationhas two dimensions, called I- and Q-branches. To reduce the demodulationcomplexity, we label each branch independently by the binary reflectedGray code (BRGC), i.e., each branch of the QAM constellation is aBRGC-labeled PAM constellation. The BRGC is generated according to thefollowing method. One example is shown in FIG. 2.

Referring to FIG. 2, an irregular BRGC labeling example for N=1, 2 and 3is shown. BRGC for N bits can be generated recursively by inserting abinary 0 or 1 to the Gray code for N-1 bits, then inserting a binary 1or 0 to the reflected (i.e., listed in reverse order) Gray code for N−1bits at the same position. The base case for N=1 bit is

$\frac{0}{1}\mspace{14mu} {or}\mspace{14mu} {\frac{1}{0}.}$

For the incoming 2N bits, c₁, c₂, . . . c_(2N), we split them into twogroups, i₁, i₂, . . . i_(2N), which is mapped to the in-phase (I)component of the QAM symbol, and q₁, q₂, . . . q_(2N), which is mappedto the quadrature component (Q). For example, for N=2, we can put the1^(st) and 4^(th) bits into the first group as i₁i₂=c₁c₄₁ and the 2^(nd)and 3^(rd) bits into the second group as q₁q₂=c₂c₃. If the 4-PAMconstellation of FIG. 2 (the 2^(nd) column) is used, i.e., {01

−3, 00

−1, 10

+1, 11

+3}, and the incoming bits are 1010, the mapped QAM symbol will be 1−j3.

In the present embodiment, we only consider the PAM constellationlabeled by the regular BRGC as an example. The regular BRGC is generatedas follows. The regular BRGC for N bits can be generated recursively byinserting a binary 1 to the Gray code for N−1 bits, then inserting abinary 0 to the reflected (i.e., listed in reverse order) Gray code forN−1 bits at the same position. The base case for N=1 bit is

$\frac{1}{0}.$

One example for N is equal to 1, 2 and 3 is shown in FIG. 3. All the QAMconstellations in the existing standards such as, e.g., WorldwideInteroperability for Microwave Access (WiMAX) and High Speed DownlinkPacket Access (HSDPA) and others (3GPP, 3GPP2, etc.) are included in thepresent method as special cases, i.e., all of them can be decomposedinto two independent PAM constellations, and each of them is labeled bythe above regular BRGC.

Referring to FIG. 3, a regular BRGC labeling example for N=1, 2 and 3 isillustratively shown.

2^(M)-PSK Constellation Construction and Modulation: In the presentembodiments, we consider the PSK constellation labeled by the inverseregular BRGC as an example. The inverse regular BRGC is generated byinverting the binary bits in FIG. 3, more accurately. The inverseregular BRGC for N bits can be generated recursively by inserting abinary 0 to the Gray code for N−1 bits, then inserting a binary 1 to thereflected (i.e., listed in reverse order) Gray code for N−1 bits at thesame position. The base case for N=1 bit is

$\frac{0}{1}.$

Referring to FIG. 4, a unit circle for the above-identified PSKconstellation is illustratively shown. We assume all the 2^(M) pointsare located on a unit circle uniformly, and number the pointscounterclockwise starting from a first point of the first quadrature.Moreover, the phase of the first point is 2^(−M)π. For example, thefirst point in the 8-PSK constellation is “000”, with phase π/8, and thelast point is “100” with phase 15π/8. It should be understood thatsimilar circles may be employed for other modulations methods (e.g.,32-PSK, PAM, QAM, etc.).

Two-Stage LLR Calculation: 2^(2N)-QAM Constellation

Hard Demodulation: As stated, the QAM constellation includes twoseparate PAM constellations. Each of these PAM constellations is labeledby the same BRGC label. Therefore, we can demodulate two branches one byone or in parallel with the same procedure. We use z to denote the realpart or the imaginary part of the received signal {tilde over (z)}, andb_(n) to denote the binary bit i_(n) or q_(n).

First, we calculate the decision metric by:

$d_{n} = \left\{ {\begin{matrix}{z,} & {n = 1} \\{{2^{N + 1 - n} - {d_{n - 1}}},} & {n \in {\left\lbrack {2,N} \right\rbrack.}}\end{matrix}.} \right.$

Then we obtain bit-level hard decisions by:

${\hat{b}}_{n} = \left\{ \begin{matrix}{0,} & {d_{n} \geq 0} & \; \\{1,} & {{{dn} < 0},} & {n \in {\left\lbrack {1,N} \right\rbrack.}}\end{matrix} \right.$

Another efficient method for symbol-level hard decisions:

$\hat{x} = \left\{ \begin{matrix}{{{{sign}(z)}\left( {2^{N} - 1} \right)},} & {{z} \geq {2^{N} - 1}} \\{{{{sign}(z)}\left\lbrack {{{FL}\left( {z} \right)} + {{LSB}\left( {{FL}\left( {z} \right)} \right)}} \right\rbrack},} & {{{z} \geq {2^{N} - 1}},}\end{matrix} \right.$

where FL(y) rounds the real number y to the nearest integer towardsminus infinity, LSB(k) is the least significant bit (LSB) of the integerk, and sign(z) gives 1 if z is greater than 0, and −1 if z is less than0.

Lookup Tables: The lookup tables for the regular BRGC-labeled 4-, 8-,16- and 32-PAM are given by Tables 1-4, respectively.

TABLE 1 The lookup table for 4-PAM constellation |{circumflex over (x)}|({circumflex over (b)}₂) ↓ (0) 3 (1) n α β α β 1 1 0 2 2 2 1 2 1 2

TABLE 2 The lookup table for 8-PAM constellation |{circumflex over (x)}|({circumflex over (b)}₂{circumflex over (b)}₃) 1 (01) 3 (00) 5 (10) 7(11) n α β α β α β α β 1 1 0 2 2 3 6 4 12 2 2 6 1 4 1 4 2 10 3 −1 −2 −1−2 1 6 1 6

TABLE 3 The lookup table for 16-PAM Constellation |{circumflex over(x)}| ({circumflex over (b)}₂{circumflex over (b)}₃{circumflex over(b)}₁) 1 (011) 3 (010) 5 (000) 7 (001) 9 (101) 11 (100) 13 (110) 15(111) n α β α β α β α β α β α β α β α β 1 1 0 2 2 3 6 4 12 5 20 6 30 742 8 56 2 4 20 3 18 2 14 1 8 1 8 2 18 3 30 4 44 3 −2 −6 −1 −4 −1 −4 −2−10 2 22 1 12 1 12 2 26 4 −1 −2 −1 −2 1 6 1 6 −1 −10 −1 −10 1 14 1 14

TABLE 4 The lookup table for 32-PAM constellation |{circumflex over(x)}| ({circumflex over (b)}₂{circumflex over (b)}₃{circumflex over(b)}₄{circumflex over (b)}₅) 1 (0111) 3 (0100) 5 (0100) 7 (0101) 9(0001) 11 (0000) 13 (0010) 15 (0011) n α β α β α β α β α β α β α β α β 11 0 2 2 3 6 4 12 5 20 6 30 7 42 8 56 2 8 72 7 70 6 66 5 60 4 52 3 42 230 1 16 3 −4 −20 −3 −18 −2 −14 −1 −8 −1 −8 −2 −18 −3 −30 −4 −44 4 −2 −6−1 −4 −1 −4 −2 −10 2 22 1 12 1 12 2 26 5 −1 −2 −1 −2 1 6 1 6 −1 −10 −1−10 1 14 1 14 |{circumflex over (x)}| ({circumflex over (b)}₂{circumflexover (b)}₃{circumflex over (b)}₄{circumflex over (b)}₅) 17 (1011) 19(1010) 21 (1000) 23 (1001) 25 (1101) 27 (1100) 29 (1110) 31 (1111) n α βα β α β α β α β α β α β α β 1 9 72 10 90 11 110 12 132 13 156 14 182 15210 16 240 2 1 16 2 34 3 54 4 76 5 100 6 126 7 154 8 184 3 4 84 3 66 246 1 24 1 24 2 50 3 78 4 108 4 −2 −38 −1 −20 −1 −20 −2 −42 2 54 1 28 128 2 58 5 −1 −18 −1 −18 1 22 1 22 −1 −26 −1 −26 1 30 1 30

Reading α and β from Tables 1-4: We can have α and β from Tables 1-4 bythe following rule, where the entry to the table is the bit position nand the last N−1 hard demodulated bits b₂ . . . b_(N):

Rule 1: (Reading α and β from a Lookup Table).

1. If {circumflex over (b)}₁=1, α and β are used as is.

2. If {circumflex over (b)}₁=0, when calculating LLR (λ₁), α is used asis, and β alternates its sign, i.e., if the value in the table is p thenβ is −p. When calculating λ_(n), n≧2, α alternates its sign and β isused as is.

Calculate bit-level LLR: Once we have α and β, we compute the LLRaccording to the following equation for each bit.

λ=4γ(αz+β)

2^(M)-PSK Constellation: Hard Demodulation:

As stated above, the incoming M bits, c₁, c₂, . . . c_(M) are mapped toa 2^(M)-PSK symbol. If the inverse regular BRGC labeling is used, thehard bits for 8-PSK are given by:

$\overset{\Cap}{c_{1}} = \left\{ {{\begin{matrix}{0,} & {{z_{Q} \geq 0},} \\{1,} & {z_{Q} < 0.}\end{matrix}\overset{\Cap}{c_{2}}} = \left\{ {{\begin{matrix}{0,} & {{z_{I} \geq 0},} \\{1,} & {z_{I} < 0.}\end{matrix}\overset{\Cap}{c_{3}}} = \left\{ \begin{matrix}{0,} & {{{z_{I}} \geq {z_{Q}}},} \\{1,} & {{z_{I}} < {{z_{Q}}.}}\end{matrix} \right.} \right.} \right.$

For 16-PSK, the 1^(st) three hard bits are decided according to theabove three equations, i.e., the decision rule for 8-PSK, and the 4^(th)hard bit is given by:

$\overset{\Cap}{c_{4}} = \left\{ \begin{matrix}{1,} & {{{\tan \; \frac{\pi}{8}} < \frac{z_{Q}}{z_{I}} < {\tan \frac{3\pi}{8}}},} \\{0,} & {{otherwise}.}\end{matrix} \right.$

For 32-PSK, the 1^(st) four hard bits are decided according to the abovefour equations, i.e., the decision rule for 16-PSK, and the 5^(th) hardbit is given by:

$\overset{\Cap}{c_{5}} = \left\{ \begin{matrix}{1,} & {{{\tan \; \frac{\pi}{16}} < \frac{z_{Q}}{z_{I}} < {\tan \; \frac{3\pi}{16}\mspace{14mu} {or}\mspace{14mu} \tan \frac{5\pi}{16}} < \frac{z_{Q}}{z_{I}} < {\tan \frac{7\pi}{16}}},} \\{0,} & {{otherwise}.}\end{matrix} \right.$

Lookup Tables: The lookup tables for the inverse regular BRGC-labeled8-, 16- and 32-PSK are given by Tables 5-7, respectively.

TABLE 5 The lookup table for 8-PSK constellation a₁a₂ a₃ k 00 01 11 10 01 1 −1 1 −1 1 0 .2706 2 1 .6533 .3827 3 1 −1 1 .2706

TABLE 6 The lookup table for 16-PSK constellation a₁a₂ a₃a₄ k 00 01 1110 00 01 11 10 1 −1 1 −1 1 0 .0747 .2126 .3928 2 1 .5879 .5133 .3753.1951 3 1 −1 1 .2126 .1379 .3182 4 .0747 −.1802

TABLE 7 The lookup table for 32-PSK constellation a₁a₂ a₃a₄a₅ k 00 01 1110 000 001 011 010 110 111 101 100 1 −1 1 −1 1 0 .0191 .0566 .1111 .1804.2619 .3525 .4486 2 1 .5466 .5275 .4900 .4355 .3622 .2847 .1942 .0980 31 −1 1 .1804 .1613 .1238 .0693 .1508 .2414 .3375 4 .0566 .0375 .0920−.1721 −.0906 −.1867 5 .0191 −.0545 .0815 −.0961

Reading K_(I) and K_(Q) from Tables 5-7: We can have K_(I) and K_(Q)from Tables 5-7 by the following rule, where the entry to the table isthe bit position k and the hard demodulated bits. In Tables 5-7, k isthe entry for bit position, a₁, a₂ is the entry for the quadrature signS, and a3 . . . am is the entry for the value K. K_(I) and K_(Q) arecalculated by the product of the quadrature sign S and the value K. Fora given hard-modulated symbol, ĉ₁ĉ₂ . . . ĉ_(m) (M is equal to 3, 4 and5 for 8-PSK, 16-PSK and 32-PSK, respectively), we have the followingrule to decide K_(I) and K_(Q).

1. For K_(I), S is read out according to the entry k=m and a₁a₂=ĉ₁ĉ₂,and K is read out according to k=m and a₃ . . . a_(m)=ĉ₃ . . . ĉ_(m).

2. For K_(Q), S is read out according to the entry k=

$\left\{ {{\begin{matrix}{1,} & {m = 2} \\{2,} & {m = 1} \\{m,} & {m \geq 3}\end{matrix}\mspace{14mu} {and}\mspace{14mu} a_{1}a_{2}} = \left\{ \begin{matrix}{{\hat{c}}_{1}{\hat{c}}_{2}} & {{\hat{c}}_{1} \neq {\hat{c}}_{2}} \\{{\overset{\overset{\_}{\hat{}}}{c}}_{1}{\overset{\overset{\_}{\hat{}}}{c}}_{\; 2}} & {otherwise}\end{matrix} \right.} \right.$

where ∘ denotes the binary inverse. In addition, S will alternate itssign if k≧4. K is read out according to k and a₃ . . . a_(m), ĉ ₃ . . .ĉ _(M).

Calculate bit-level LLR: Once we have K_(I) and K_(Q) we compute the LLRaccording to the following equation for each bit.

λ=4γ(K ₁ z ₁ +K _(Q) z _(Q))

Although the above hard demodulation and lookup tables are derived forregular BRGC-labeled QAM and inverse regular BRGC-labeled PSKconstellations, we want to emphasize the above idea can be easilyapplied to other BRGC-labeled QAM and PSK constellations. This isbecause the lookup table will be fixed when the constellation is given.

Referring to FIG. 5, a block/flow diagram showing a system/method fordemodulation is illustratively depicted in accordance with the presentprinciples. In block 102, a received signal is demodulated to obtainhard bit or hard symbol information. The demodulating includes mappingthe incoming information to hard bits in accordance with a 2^(2N)quadrature amplitude modulation (QAM) constellation (path 120) or inaccordance with a 2^(M) pulse shift keying modulation (PSK)constellation (path 130).

Along path 120, in block 104, the QAM constellation includes twobranches, and the two branches each employ a pulse amplitude modulation(PAM) constellation for mapping the hard bits. In block 106, each branchis labeled with the same labeling method and preferably in accordancewith a regular binary reflected Gray code (BRGC). Along path 130, inblock 103, the PSK constellation is labeled preferably in accordancewith an inverse regular binary reflected Gray code (BRGC).

In block 108, coefficients from at least one lookup table are referencedand obtained to compute log-likelihood ratios (LLRs) from the hard bitor symbol information.

In path 120, in block 110, computing bit-level log-likelihood ratiosincludes computing the LLRs based upon coefficients α and β obtainedfrom the at least one lookup table and an additive noise coefficientcomputed from the received signal, wherein the coefficients α and β areindexed by bit/symbol estimates and bit position.

In path 130, in block 111, computing bit-level log-likelihood ratiosincludes computing the LLRs based upon coefficients K_(I) and K_(Q)obtained from the at least one lookup table and an additive noisecoefficient computed from the received signal, wherein the coefficientsK_(I) and K_(Q) are indexed by hard modulated bits and bit position. Thehard modulated bits provide a quadrature sign (S) and a value K whichare employed to lookup K_(I) and K_(Q).

In block 112, bit-level log-likelihood ratios LLRs are computed from thecoefficients and the received signal. LLRs are employed in bitpredictions and are employed as is known in the art.

Having described preferred embodiments for a two-stage low-complexitymax-log bit-level LLR calculator and method (which are intended to beillustrative and not limiting), it is noted that modifications andvariations can be made by persons skilled in the art in light of theabove teachings. It is therefore to be understood that changes may bemade in the particular embodiments disclosed which are within the scopeand spirit of the invention as outlined by the appended claims. Havingthus described aspects of the invention, with the details andparticularity required by the patent laws, what is claimed and desiredprotected by Letters Patent is set forth in the appended claims.

1. A demodulator comprising: a bit/symbol hard demodulator configured toobtain hard bit or symbol information from a received signal; at leastone lookup table configured to reference coefficients for computation oflog-likelihood ratios (LLRs) from the hard bit or symbol information;and a log-likelihood ratio calculation module configured to computebit-level LLRs from the coefficients and the received signal.
 2. Thedemodulator as recited in claim 1, wherein the received signal includesin-phase and quadrature components and the hard demodulators furthercomprising a 2^(2N) quadrature amplitude modulation (QAM) constellationfor mapping the incoming information to hard bits.
 3. The demodulator asrecited in claim 2, wherein the QAM constellation includes two branches,and the two branches each include a pulse amplitude modulation (PAM)constellation.
 4. The demodulator as recited in claim 3, wherein eachbranch is labeled in accordance with a binary reflected Gray code(BRGC).
 5. The demodulator as recited in claim 2, wherein log-likelihoodratio calculation module computes the LLR based upon coefficients α andβ obtained from the at least one lookup table and an additive noisecoefficient computed from the received signal.
 6. The demodulator asrecited in claim 5, wherein the coefficients α and β are indexed by bitand symbol estimates and bit position.
 7. The demodulator as recited inclaim 1, wherein the received signal includes in-phase and quadraturecomponents and the hard demodulator further comprising a 2^(M) pulseshift keying modulation (PSK) constellation for mapping the incominginformation to hard bits.
 8. The demodulator as recited in claim 7,wherein PSK constellation is labeled in accordance with an inverseregular binary reflected Gray code (BRGC).
 9. The demodulator as recitedin claim 7, wherein log-likelihood ratio calculation module computes theLLR based upon coefficients K_(I) and K_(Q) obtained from the at leastone lookup table and an additive noise coefficient computed from thereceived signal.
 10. The demodulator as recited in claim 9, wherein thecoefficients K_(I) and K_(Q) are indexed by hard modulated bits and bitposition.
 11. The demodulator as recited in claim 10, wherein the hardmodulated bits provide a quadrature sign (S) and a value K which areemployed to lookup K_(I) and K_(Q).
 12. A method for demodulation,comprising: demodulating a received signal to obtain hard bit or symbolinformation; referencing coefficients from at least one lookup table tocompute log-likelihood ratios (LLRs) from the hard bit or symbolinformation; computing bit-level log-likelihood ratios LLRs from thecoefficients and the received signal.
 13. The method as recited in claim12, wherein demodulating includes mapping the incoming information tohard bits in accordance with a 2^(2N) quadrature amplitude modulation(QAM) constellation.
 14. The method as recited in claim 13, wherein theQAM constellation includes two branches, and the two branches eachemploy a pulse amplitude modulation (PAM) constellation for mapping thehard bits.
 15. The method as recited in claim 14, further comprisinglabeling each branch in accordance with a binary reflected Gray code(BRGC).
 16. The method as recited in claim 13, wherein computingbit-level log-likelihood ratios includes computing the LLRs based uponcoefficients α and α obtained from the at least one lookup table and anadditive noise coefficient computed from the received signal, whereinthe coefficients α and β are indexed by bit/symbol estimates and bitposition.
 17. The method as recited in claim 12, wherein demodulatingincludes mapping the incoming information to hard bits in accordancewith a 2^(M) pulse shift keying modulation (PSK) constellation.
 18. Themethod as recited in claim 17, further comprising labeling the PSKconstellation in accordance with an inverse regular binary reflectedGray code (BRGC).
 19. The method as recited in claim 17, whereincomputing bit-level log-likelihood ratios includes computing the LLRsbased upon coefficients K_(I) and K_(Q) obtained from the at least onelookup table and an additive noise coefficient computed from thereceived signal, wherein the coefficients K_(I) and K_(Q) are indexed byhard modulated bits and bit position.
 20. The method as recited in claim19, wherein the hard modulated bits provide a quadrature sign (S) and avalue K which are employed to lookup K_(I) and K_(Q).